Cooperative Source Seeking for Uncertain Networked Euler–Lagrange Systems over Unbalanced Digraphs

This work investigates the cooperative source seeking problem for uncertain networked Euler–Lagrange (EL) systems over unbalanced digraphs. The objective is to drive all agents towards the source of scalar fields. To achieve this, a novel distributed control protocol is proposed, which consists of two components: a distributed optimizer that generates a virtual reference trajectory towards the source, and a trajectory tracker that drives each agent exponentially towards this trajectory. Complemented by the exponential trajectory tracker, the stochastic extremum seeking techniques can be employed to estimate the gradient information from the strength of the scalar field in the optimizer. However, this approach introduces a new coupling term to the optimizer at the same time, which requires careful consideration in the convergence analysis. Leveraging stochastic averaging theory, the exponential practical stability then can be guaranteed for the entire closed-loop system. Finally, the effectiveness of the proposed control protocol is demonstrated by a numerical example.